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Theorem irredrmul 20359

Description: The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)

**Hypotheses**
Ref	Expression
irredn0.i	⊢ 𝐼 = (Irred‘𝑅)
irredrmul.u	⊢ 𝑈 = (Unit‘𝑅)
irredrmul.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
irredrmul	⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ 𝐼)

Proof of Theorem irredrmul Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1135	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐼)
2		simp1 1134	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → 𝑅 ∈ Ring)
3		simp3 1136	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈)
4		irredrmul.u	. . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑅)
5		eqid 2728	. . . . . . . . 9 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
6	4, 5	unitdvcl 20337	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌)(/_r‘𝑅)𝑌) ∈ 𝑈)
7	6	3com23 1124	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈 ∧ (𝑋 · 𝑌) ∈ 𝑈) → ((𝑋 · 𝑌)(/_r‘𝑅)𝑌) ∈ 𝑈)
8	7	3expia 1119	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) ∈ 𝑈 → ((𝑋 · 𝑌)(/_r‘𝑅)𝑌) ∈ 𝑈))
9	2, 3, 8	syl2anc 583	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) ∈ 𝑈 → ((𝑋 · 𝑌)(/_r‘𝑅)𝑌) ∈ 𝑈))
10		irredn0.i	. . . . . . . . 9 ⊢ 𝐼 = (Irred‘𝑅)
11		eqid 2728	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	irredcl 20356	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ (Base‘𝑅))
13	12	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅))
14		irredrmul.t	. . . . . . . 8 ⊢ · = (._r‘𝑅)
15	11, 4, 5, 14	dvrcan3 20342	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌)(/_r‘𝑅)𝑌) = 𝑋)
16	2, 13, 3, 15	syl3anc 1369	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌)(/_r‘𝑅)𝑌) = 𝑋)
17	16	eleq1d 2814	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (((𝑋 · 𝑌)(/_r‘𝑅)𝑌) ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
18	9, 17	sylibd 238	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) ∈ 𝑈 → 𝑋 ∈ 𝑈))
19	2	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → 𝑅 ∈ Ring)
20		eldifi 4122	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) → 𝑦 ∈ (Base‘𝑅))
21	20	ad2antrl 727	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → 𝑦 ∈ (Base‘𝑅))
22	3	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → 𝑌 ∈ 𝑈)
23	11, 4, 5	dvrcl 20336	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑌 ∈ 𝑈) → (𝑦(/_r‘𝑅)𝑌) ∈ (Base‘𝑅))
24	19, 21, 22, 23	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → (𝑦(/_r‘𝑅)𝑌) ∈ (Base‘𝑅))
25		eldifn 4123	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) → ¬ 𝑦 ∈ 𝑈)
26	25	ad2antrl 727	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ¬ 𝑦 ∈ 𝑈)
27	4, 14	unitmulcl 20312	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑦(/_r‘𝑅)𝑌) ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → ((𝑦(/_r‘𝑅)𝑌) · 𝑌) ∈ 𝑈)
28	27	3com23 1124	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈 ∧ (𝑦(/_r‘𝑅)𝑌) ∈ 𝑈) → ((𝑦(/_r‘𝑅)𝑌) · 𝑌) ∈ 𝑈)
29	28	3expia 1119	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → ((𝑦(/_r‘𝑅)𝑌) ∈ 𝑈 → ((𝑦(/_r‘𝑅)𝑌) · 𝑌) ∈ 𝑈))
30	19, 22, 29	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ((𝑦(/_r‘𝑅)𝑌) ∈ 𝑈 → ((𝑦(/_r‘𝑅)𝑌) · 𝑌) ∈ 𝑈))
31	11, 4, 5, 14	dvrcan1 20341	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑌 ∈ 𝑈) → ((𝑦(/_r‘𝑅)𝑌) · 𝑌) = 𝑦)
32	19, 21, 22, 31	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ((𝑦(/_r‘𝑅)𝑌) · 𝑌) = 𝑦)
33	32	eleq1d 2814	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → (((𝑦(/_r‘𝑅)𝑌) · 𝑌) ∈ 𝑈 ↔ 𝑦 ∈ 𝑈))
34	30, 33	sylibd 238	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ((𝑦(/_r‘𝑅)𝑌) ∈ 𝑈 → 𝑦 ∈ 𝑈))
35	26, 34	mtod 197	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ¬ (𝑦(/_r‘𝑅)𝑌) ∈ 𝑈)
36	24, 35	eldifd 3956	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → (𝑦(/_r‘𝑅)𝑌) ∈ ((Base‘𝑅) ∖ 𝑈))
37		simprr 772	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → (𝑥 · 𝑦) = (𝑋 · 𝑌))
38	37	oveq1d 7429	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ((𝑥 · 𝑦)(/_r‘𝑅)𝑌) = ((𝑋 · 𝑌)(/_r‘𝑅)𝑌))
39		eldifi 4122	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝑅) ∖ 𝑈) → 𝑥 ∈ (Base‘𝑅))
40	39	ad2antlr 726	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → 𝑥 ∈ (Base‘𝑅))
41	11, 4, 5, 14	dvrass 20340	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑌 ∈ 𝑈)) → ((𝑥 · 𝑦)(/_r‘𝑅)𝑌) = (𝑥 · (𝑦(/_r‘𝑅)𝑌)))
42	19, 40, 21, 22, 41	syl13anc 1370	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ((𝑥 · 𝑦)(/_r‘𝑅)𝑌) = (𝑥 · (𝑦(/_r‘𝑅)𝑌)))
43	16	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ((𝑋 · 𝑌)(/_r‘𝑅)𝑌) = 𝑋)
44	38, 42, 43	3eqtr3d 2776	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → (𝑥 · (𝑦(/_r‘𝑅)𝑌)) = 𝑋)
45		oveq2 7422	. . . . . . . . 9 ⊢ (𝑧 = (𝑦(/_r‘𝑅)𝑌) → (𝑥 · 𝑧) = (𝑥 · (𝑦(/_r‘𝑅)𝑌)))
46	45	eqeq1d 2730	. . . . . . . 8 ⊢ (𝑧 = (𝑦(/_r‘𝑅)𝑌) → ((𝑥 · 𝑧) = 𝑋 ↔ (𝑥 · (𝑦(/_r‘𝑅)𝑌)) = 𝑋))
47	46	rspcev 3608	. . . . . . 7 ⊢ (((𝑦(/_r‘𝑅)𝑌) ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · (𝑦(/_r‘𝑅)𝑌)) = 𝑋) → ∃𝑧 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑧) = 𝑋)
48	36, 44, 47	syl2anc 583	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) ∧ (𝑦 ∈ ((Base‘𝑅) ∖ 𝑈) ∧ (𝑥 · 𝑦) = (𝑋 · 𝑌))) → ∃𝑧 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑧) = 𝑋)
49	48	rexlimdvaa 3152	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)) → (∃𝑦 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑦) = (𝑋 · 𝑌) → ∃𝑧 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑧) = 𝑋))
50	49	reximdva 3164	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑦 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑦) = (𝑋 · 𝑌) → ∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑧 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑧) = 𝑋))
51	18, 50	orim12d 963	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (((𝑋 · 𝑌) ∈ 𝑈 ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑦 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑦) = (𝑋 · 𝑌)) → (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑧 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑧) = 𝑋)))
52	11, 4	unitcl 20307	. . . . . 6 ⊢ (𝑌 ∈ 𝑈 → 𝑌 ∈ (Base‘𝑅))
53	52	3ad2ant3 1133	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Base‘𝑅))
54	11, 14	ringcl 20183	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ 𝑌 ∈ (Base‘𝑅)) → (𝑋 · 𝑌) ∈ (Base‘𝑅))
55	2, 13, 53, 54	syl3anc 1369	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ (Base‘𝑅))
56		eqid 2728	. . . . 5 ⊢ ((Base‘𝑅) ∖ 𝑈) = ((Base‘𝑅) ∖ 𝑈)
57	11, 4, 10, 56, 14	isnirred 20352	. . . 4 ⊢ ((𝑋 · 𝑌) ∈ (Base‘𝑅) → (¬ (𝑋 · 𝑌) ∈ 𝐼 ↔ ((𝑋 · 𝑌) ∈ 𝑈 ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑦 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑦) = (𝑋 · 𝑌))))
58	55, 57	syl 17	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (¬ (𝑋 · 𝑌) ∈ 𝐼 ↔ ((𝑋 · 𝑌) ∈ 𝑈 ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑦 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑦) = (𝑋 · 𝑌))))
59	11, 4, 10, 56, 14	isnirred 20352	. . . 4 ⊢ (𝑋 ∈ (Base‘𝑅) → (¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑧 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑧) = 𝑋)))
60	13, 59	syl 17	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ 𝑈)∃𝑧 ∈ ((Base‘𝑅) ∖ 𝑈)(𝑥 · 𝑧) = 𝑋)))
61	51, 58, 60	3imtr4d 294	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (¬ (𝑋 · 𝑌) ∈ 𝐼 → ¬ 𝑋 ∈ 𝐼))
62	1, 61	mt4d 117	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3066 ∖ cdif 3942 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 ._rcmulr 17227 Ringcrg 20166 Unitcui 20287 Irredcir 20288 /_rcdvr 20332

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-irred 20291 df-invr 20320 df-dvr 20333

This theorem is referenced by: irredlmul 20360 irredneg 20362

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